Enhanced directionality of active processes in a viscoelastic bath

Active fluctuations are known to play a significant role in the intracellular transport of passive objects. However, the effect of viscoelasticity of the environment in shaping such processes is relatively less understood. Here, with a minimal experiment using a driven colloid in a viscoelastic bath, we show that viscoelasticity significantly increases the mean injected power to the passive object ($\sim 50\%$ compared to a viscous medium), for the same strength of the external driving. Additionally, we observe a notable reduction in negative work fluctuations across a wide range of driving amplitudes. These findings collectively suggest an enhanced directionality in driven processes within a viscoelastic bath, which we attribute to the emergence of interactions between the colloid and the viscoelastic medium.


Introduction
Microscopic biophysical processes are vital for the functioning of living organisms.Many of these processes, such as intracellular transport of passive objects [1,2], are non-equilibrium in nature, mostly driven by ATP-consuming molecular motors such as kinesins [3,4].Developing accessible models for these processes is of utmost importance as it provides novel perspectives into the functioning of the fundamental components of life within complex environments.
In a zeroth order approximation, such processes can be modelled assuming a separation of time-scales between the environmental and system degrees of freedom, leading to, effectively, a Markovian description for the dynamics of the system.The thermodynamics of such processes has been extensively studied in recent decades, particularly in the context of stochastic thermodynamics [5,6].This field has focused on various aspects, including fluctuation relations [7,8,9], as well as advancements in stochastic inference techniques [10,11,12,13,14,15,16,17,18].These theories have now been well tested in experiments, and have provided us with accurate estimates of thermodynamic quantities, such as equilibrium free energy differences and dissipation for specific biophysical systems [19,20] and efficiencies of molecular motors [21,22].
A vast majority of these experiments are carried out in in-vitro conditions where the experimental medium is usually a homogeneous viscous solution and the Markovian assumption is valid.This is also the case for most artificially synthesized molecular motors, Brownian Carnot engines and other nanoscale devices [22,23].Contrary to this, in reality, microscopic biophysical processes differ from simplified models by two key features.Firstly, living cells possess inherent activity and dynamic behavior, contributing to complex fluctuations that primarily govern the underlying processes, except in some situations where passive diffusion also may play big roles.For example, diffusive processes with directed motility of various intracellular elements are crucial in positioning and distributing organelles inside a cell [24,25].However, overall, the hidden nature of many such sub-cellular processes leads to non-Markovian dynamics for the observable microscopic processes.Secondly, the intracellular medium is highly viscoelastic.This viscoelasticity arises from the intricate composition and structure of the cellular environment, characterized by a combination of viscous and elastic properties [26,27].This also leads to memory effects due to the large relaxation times, which again breaks the effective Markovian description for the dynamics of the system [28,29,30,31,32,33].
Quite naturally, both of these issues have individually generated significant recent interest.Studies on various systems, such as bacterial suspensions or self-propelled particles, have revealed that active fluctuations can induce non-trivial transport phenomena inside cells [34,35], enhance mixing [36] and diffusion [37,38], and even drive the emergence of collective behaviour [39].Studies have also shown that active fluctuations can drive the self-assembly of colloidal systems [40], lead to fluctuation induced phase transitions [41], and enhance the efficiency of microscopic heat engines [42].Yet another interesting observation was made in Ref. [43], which showed that the molecular motor kinesin accelerates in the presence of an active noise in the environment, implying active fluctuations in cells are utilized to promote various physiological processes.This finding was further explored in detail recently in Ref. [44].
Similarly, recent experiments have shown several intriguing phenomena related to the dynamics of an embedded particle as well as swimmers in viscoelastic fluids [45,46].These observations include the increase in Kramer's rate of transitions [47,48,49], circular motion of active particles instead of only persistent random walks [50,51], increase in the rotational diffusion coefficient of an active Brownian particle by orders of magnitude compared to that in a viscous fluid [51], and very interestingly, the possibility of attaining the Carnot efficiency in microscopic sterling engine [52].Several recent studies also show that viscoelasticity is essential for facilitating various biological processes [53,35,54].
Compared to the vast literature on the individual fields, the study of the combined effects of active processes in viscoelastic baths is relatively underexplored.Some notable findings in this emerging area include the swelling of a polymer in an active and viscoelastic bath, where the extent of swelling depends on the viscoelastic properties of the environment [55], the study of active diffusion of model chromosomal loci driven by athermal noise, revealing the influence of the polymeric effect on the anomalous diffusion exponent [56], the reduction of quasi-static efficiency in Brownian engines due to the viscoelasticity of the active suspension [57], and the implications of complex multi-scale effects leading to time-dependent temperature [58].
Despite these recent progresses, two interconnected open questions remain: Does the viscoelasticity of the environment affect the performance of micro-engines driven by active processes, and does it influence the arrow of time and fluctuations associated with such processes?In this work, we address these issues through a minimal experiment in a viscoelastic environment.We consider a colloidal particle in a harmonic trap, whose mean position is modulated according to an Ornstein-Uhlenbeck (OU) process.The usage of this type of driving is very common in the context of active matter systems, where the OU process is typically used to mimic the interaction of the passive colloid with the surrounding active particles (such as molecular motors) [59,60,57,61].The stochastic thermodynamics of this system in a viscous medium has been extensively studied both experimentally [62,14,63] and theoretically [64,65,66,67,68,60].
In this paper, we study this model in a homogeneous viscoelastic medium, for which there are no known results to the best of our knowledge.We first analyze the particle dynamics experimentally and theoretically using a generalised Langevin model.The model is solved analytically and the solutions match well with our experimental results.We then proceed to characterizing the non-equilibrium thermodynamics of the system.We calculate the average power supplied to the particle by stochastic driving, which in this case is the same as the thermodynamic arrow of time or total entropy production of the process, up to a scaling factor of the temperature of the environment.Interestingly, coupling the system with a viscoelastic medium, as opposed to a viscous medium with the same zero-shear viscosity, leads to approximately 50% enhancement in the mean value of this quantity.This suggests that, rather counter intuitively, viscoelasticity amplifies the process's directionality under identical external driving forces.Furthermore, based on a recent theory for interacting particle systems, we propose that this increase in entropy production in a viscoelastic bath is a generic property, independent of the system's specifics.Next, we look into the fluctuations of the injected power both experimentally and theoretically.In particular, we obtain previously unknown, exact analytical solutions for the work distribution at shorttimes, which are non-Gaussian, yet completely determined in terms of the mean and variance of the single-step work fluctuations.The solution obeys a modified fluctuation relation P (W )/P (−W ) = exp(−2β ws W ), where 2β ws ≪ 1.We also obtain an approximate solution for arbitrary time work distribution, which agree excellently with the experimental data.Interestingly, we observe a significant reduction in negative work fluctuations, both at short and arbitrary times, in the viscoelastic environment compared to their viscous counterparts.This finding is consistent with the improved arrow of time we observed, highlighting the influence of viscoelasticity on the directionality of the process.

Results
Our experimental system primarily consists of an optically trapped, spherical polystyrene particle of diameter 2a 0 = 3 µm (Sigma Aldrich) dispersed in a 0.05% (w/v) aqueous solution of poly-ethylene oxide (PEO) [Fig.1(a)].PEO is a polymer that creates a stable configuration in water, which provides elasticity along with its inherent viscosity and renders the solution viscoelastic.The stiffness of the trap is measured to be 12 ± 0.5 pN/µm.To mimic the active noise in the background of different strengths, we spatially modulate the trapping laser using an acousto-optic modulator (AOM) connected to a signal generator that generates the Orstein-Uhlenbeck (OU) noise of various amplitudes.The trajectory we measure consists of the position of the laser spot x 0 (t), as well as the one-dimensional position of the particle x(t) at a spatio-temporal resolution of 1nm − 10kHz.The readings are obtained using a balanced detector for ∼ 100s (see Appendix E for details).
It is known that x(t) is well described using a generalized stochastic differential equation [69]: Here the function Γ(t) is referred to as the memory kernel.Further, the noise term η ′ (t) satisfied ⟨η ′ (t)⟩ = 0, and has correlations ⟨η where k B is the Boltzmann constant and T is the temperature of the bath [69].We further modulate x 0 (t) according to the Ornstein-Uhlenbeck process: Here ϕ ext (t) is a zero-mean delta-correlated Gaussian noise with standard deviation equal to 1, i.e., ⟨ϕ ext (t A is the strength of that noise (which we experimentally vary using the AOM), and τ e is the correlation time of that process.The correlation of x 0 is given by ⟨x 0 (t)x 0 (t For the memory kernel Γ, we consider the Jeffrey's fluid model of viscoelasticity [30,50,51,47,52].According to this model [69,70,71,72]: Here τ is the relaxation time of the fluid, γ 0 and γ are the solvent and the polymer contributions of the frictional coeffecient, respectively [70].Notably, the memory kernel will be viscous-like (delta-correlated) in the τ → 0 limit as discussed in Refs.[47,52].
Hereafter, we refer to this limit as the viscous limit.There are two distinct advantages of using this model for the viscoelastic bath.Firstly, we can identify the parameters that appear in this model using independent measurements.One such mapping is due to a technique developed in Ref. [32].In this approach, one measures the phase response of a particle embedded in the fluid to an external perturbation and computes the relevant parameters by fitting it to theoretical predictions obtained using the Jeffrey's fluid model.By this method, the phase response of the particle over a wide frequency range can be obtained from a single trajectory using a 3 Hz square wave as the perturbation (For the details, see Appendix D).We perform this analysis in our setup and the results are shown in Fig. D1 of Appendix D. We obtain the estimates τ = 0.018 ± 0.002s and the ratio between the polymer contribution to the viscosity and solvent contribution to the viscosity, µ r (≡ γ/γ 0 ) = 1.805 ± 0.10.Secondly, as we show in Appendix AI, using the values of parameters inferred in the previous step, and by introducing an additional auxiliary variable X(t), we can write down a Markovian evolution equation in a three dimensional space, whose projection to one dimension results in the Langevin equation in Eq. ( 1).The same approach has also been used previously in other works to convert a non-Markovian process into Markovian so that the dynamics may be determined conveniently [69,72].Moreover, it is to be noted that this method can also be extended easily to other more general viscoelastic Maxwellian fluids with more than one relaxation time [73].
Introducing the auxiliary variable X(t), the total dynamical system can then be described using a three-dimensional, linear, stochastic differential equation, Where Y (t) = [x(t), X(t), x 0 (t)] T , and, Here . For the convenience of our discussion, we further define a dimensionless parameter θ As we keep all other parameters except A fixed in our experiment, the variation of θ directly maps to the variation of A (i.e., the strength of the OU noise).
Eq. ( 4) is an example of a linear multivariate OU process, and the corresponding Fokker-Planck equation can be solved using standard techniques [74,69] to compute the steady state distribution P ss (Y ) (see the Appendix AI).Due to the presence of the Ornstein-Uhlenbeck driving, this steady state will be not be in equilibrium, being characterized by a nonzero probability flux J ss (Y ), as shown in Fig. 1(b).
Using this analytical solution, we can further obtain the variance of the particle position as, Note that we can obtain the variance of P (x) for an active viscous bath (Eq.(A.12) of Appendix AI) by considering the limit τ → 0 in Eq. ( 6) which is exactly similar to the related expression given in Ref. [59].
Our experimental measurements agree very well with the model.In Fig. 2(a), we show the experimentally measured mean-square displacement functions (MSD) along with the corresponding theoretical lines for two different amplitudes of the OU noise.We also show the theoretical MSD line in grey colour for zero external applied noise.Each of these MSD flattens at large times due to the spatial confinement of the particle by optical tweezers, and the corresponding value depends linearly on the effective temperature (defined from the variance of the probability distribution function as In the θ → 0 limit, it reaches the equilibrium value 2k B T /k.In Fig. 2(b) we show experimental measures of MSDs in the viscoelastic and viscous fluids for a fixed value of θ along with corresponding theory curves.For experiments in the viscous fluid, we have chosen a solution of ultra-pure glycerol in water with volume fraction ∼ 0.254 to reach the viscosity ∼ 2452 µP.s at the room temperature 20 • C, which is very close to the zeroshear viscosity (when τ → 0) of the viscoelastic fluid, i.e., 2496 µP.s = (γ 0 + γ)/6πa 0 [75,76].Note that due to the change in the refractive index when glycerol is mixed in water, the trap stiffness gets changed.To bring the stiffness back to our fixed value, we changed the trapping laser power appropriately.
It is clear in Fig. 2(b) that the effective temperature in a viscoelastic fluid is slightly higher as compared to that in a viscous fluid.Further, due to the change in the fluid time scale, a prominent difference in the short-time region is also visible.Fig. 2(c) represents the measured steady-state probability distributions of x(t) for the same two noise amplitudes as in Fig. 2(a) in the viscoelastic fluid.Certainly, due to higher effective temperature for θ = 0.56, the broadening of the corresponding distribution is higher compared to that of θ = 0.12.Interestingly, the higher effective temperature is also visible in Fig. 2(d) which shows the experimental and theoretical steady-state probability distribution of x(t) in viscous and viscoelastic fluids, for a fixed value of θ.
In Fig. 2(e), we plot Var(x(t)) as a function of τ .We find that for large values of τ , Var(x(t)) saturates to a constant value, which is higher than the corresponding value in the viscous (τ → 0) limit.Interestingly, however, Var(x(t)) is non-monotonic in τ and features a minimum at an intermediate τ value, that depends on the two other inherent time-scales of the system only and not on θ (see Eq.(A.11) of the Appendix AI for the expression of τ corresponding to the minima).We show this in the inset of Fig. 2(e).Note that this is purely an effect of viscoelasticity of the medium since the effect of the OU driving alone does not result in this non-monotonic behaviour (see Ref. [59]).
Now we move on to study the non-equilibrium thermodynamics of this system.To this end, we look at the statistics of the injected mechanical work done by the external forcing on the colloidal particle, defined as [77,6,65,66,63]: Using this expression, the average power delivered to the particle can be straightforwardly computed from the experimental data both in viscous and viscoelastic fluids.The results are shown in Fig. 3(a), as a function of noise-strength θ.We find that ⟨W τm ⟩ is positive as expected from the Second Law, and increases linearly as a function of θ.The estimates also agree well with the analytically computed estimate (solid lines in Fig. 3(a), see Appendix AII for details) from the model, given by, Interestingly, the obtained average power in a viscoelastic bath is significantly higher compared to the same for a 'memory-less' Newtonian bath with our chosen parameters.This implies that for the same amplitude of external driving, a higher power is delivered to a particle in a viscoelastic bath compared to its viscous counterpart (See the Eq.(A.17) and Eq.(A.18) of Appendix AII for corresponding expressions in terms of θ both for viscous and viscoelastic fluids for a comparison).Even though our finding is for a particular example of a non-equilibrium system, it is possible to argue that this is a generically expected property for any non-equilibrium system in a viscoelastic environment.The argument is based on the observation that the average injected power for this system is the same as the average total entropy production, irrespective of the rheological properties of the bath (See Appendix C).This is because the injected power is the only time extensive source of dissipation in the system.Furthermore, for systems in contact with a heat reservoir, this quantity can be considered as representative of the global arrow of time, defined as, where Y (•) = {Y (t)} T t=0 is the trajectory of the system measured for a time duration T , and Ỹ (t) is its time reversed counterpart defined as Ỹ (t) = Y (T − t).It was recently shown that the presence of interactions increases the evidence of the global arrow of time [78].In a viscoelastic bath, we know already that the polymers in the viscoelastic solution contribute to additional interactions and that can significantly affect the statistics of the position of the particle.Thus we argue that the increase in entropy production is due to the interaction of the colloidal particle with polymers in the solution.We note that a similar argument can be put forward on the basis of the theoretical model, where the random variable X(t) is the additional interacting degree of freedom which is absent in a viscous bath.
A natural question that then arises is whether the arrow of time of the process can be continuously increased by working with a medium of increased viscoelastic response.
Although this is experimentally challenging to address, we can seek an answer using the expressions derived using the theoretical model.In Fig. 3(b), we show that in the limit of increasing relaxation time τ of the viscoelastic bath, ⟨W τm ⟩ increases and saturates to a constant value that depends only on the properties of the trap parameters.However, in contrast to the non-monotonic behavior observed for Var(x(t)), we find that ⟨W τm ⟩ exhibits a monotonically increasing function of τ .This observation reinforces the suggestion that hidden interactions play a significant role in the amplification of the thermodynamic arrow of time.
Finally, we look at the fluctuations of the injected power by computing the probability distribution P (W τm ).In certain cases, it is known that this probability distribution obeys a fluctuation theorem of the form: where τ l denotes the largest time scale of the system and k B is the Boltzmann constant.The work fluctuation theorem has already been tested and validated in diverse experiments and numerical simulations.Similar setups to ours include an optically trapped particle dragged with a constant velocity [79], Brownian oscillator with deterministic periodic driving in the non-markovian bath [80], a colloidal particle trapped in a bi-stable potential subjected to periodic modulation [81], Brownian particle trapped in an optical trap with discontinuously varying trap stiffness [82,83] and also in other cases such as electronic circuits [84], mechanical oscillators [85], etc.However, violations are also found in many systems, e.g., a colloidal particle under the action of a Gaussian distributed white and coloured noise [86,87,62,68], wave turbulence [88], an RC circuit driven with a stochastic voltage [89], a rotating active Brownian particle in a viscoelastic medium [90], etc.These violations have been explored very well, and it is known that the primary reason for the violation is the presence of time intensive contributions to the total entropy production, which can significantly contribute to fluctuations at short times and in continuous state space systems [91,92].An exact expression for the work probability distribution that appears in the fluctuation relations is known only in a handful of cases.For example, in the case of a colloidal particle in a harmonic trap with the mean position externally controlled, for any arbitrary deterministic driving, the work distribution can be shown to be a Gaussian [93,94].The case where the stiffness is time dependently controlled is one of the simplest models showing non-Gaussian work fluctuations [95].Another system for which an exact calculation of the work distribution has been carried out is the colloidal particle in a logarithmic-harmonic potential [96], which also has non-Gaussian work fluctuations.A similar occurrence of non-Gaussian work fluctuations in a complex fluid in the context of an optically trapped particle with deterministically varying trap stiffness has also been discussed by Carberry et.al. in Ref. [83].In [97], exact work statistics have also been obtained for a colloidal particle in the uniformly dragged two dimensional confining  potential, and in the presence of additional non-conservative forces such as torques.In this case, however, the work distribution is found to be a Gaussian, although it deviates from the conventional work fluctuation theorem.A general framework for computing the large-time form of the work distribution, for linear Langevin processes, is described in Refs.[80,64,65].The method is based on the calculation of the corresponding moment-generating function using a large deviation technique and gives closed form expressions exact to sub leading order in 1/t.The methods developed in Refs.[66,67] provides exact analytical expressions for the moment generating function for any value of t for the same class of systems.However, inverting them to obtain a closed form expression for P (W, t) is challenging.
Here, using a novel technique, we are able to compute a closed form expression of the probability distribution P (W τm ) in the single-step (short-time) limit.This limit has received a lot of attention recently, primarily in the context of saturation of the thermodynamic uncertainty relation [11,13,12,14,15,98,99].In particular, it is known that in the strict limit of τ m → 0 fluctuations of any time-integrated current in an overdamped diffusive system is Gaussian.The single-step limit we consider here is the closest to the short-time limit (that is experimentally accessible), where the distribution is expected to be non-Gaussian.The technique we use is based on the fact that the singlestep work W appears as the product of two zero-mean Gaussian distributed quantities ẋ(t) and x 0 (t) which are cross-correlated (see Eq. ( 7)).The probability distribution of W , therefore, can be obtained directly from the joint probability distribution function of ẋ(t) and x 0 (t) as a skewed zeroth-order modified Bessel function [100] where , The subscript 'ws' stands for single-step work.See Appendix AIII for a detailed derivation of the expression above.
Clearly, the expression above is non-Gaussian.However, interestingly, it is entirely determined in terms of the mean and variance of single-step work.Further note that, except for an exponential factor of e βwsW , the distribution P (W ) does not depend on the sign of W . Hence, it is straightforward to write down a modified fluctuation relation, of the form: In Fig. 4 (a) -(d) and (i) -(l), we show the excellent agreement between this analytically computed expression with the experimental as well as numerical data.In Fig. 4 (e) -(h), and (m) -(p) we again verify this fluctuation theorem using both experimental and numerical data.By definition, β ws corresponds to the relative likelihood of having a positive work fluctuation over the negative one of the same magnitude.It is hence interesting to check how this factor is affected by the viscoelasticity of the medium.In Fig. 5, we plot β ws as a function of the viscoelastic fluid time-scale τ and find that it monotonically increases with it.This trend is qualitatively similar to that observed for the average entropy production rate (see Fig. 3b), and shows that for a fixed strength of non-equilibrium driving, a positive work fluctuation is relatively more favoured in a viscoelastic bath with a higher value of τ .We have further found that the effect of viscoelasticity on β ws is much more significant as compared to the effect of θ on the same.In the inset of Fig. 5, we analytically compare β ws obtained for a range of θ values for the viscoelastic case and viscous case.We find that β ws for the viscoelastic case is always higher than β ws for the viscous case, but in both cases, it varies negligibly with θ.
Next, we consider finite time work distributions, which represent the distributions of cumulative work W τm obtained using Eq.(7).To approximate this distribution, we assume an uncorrelated time series of short-time work and utilize the probability distribution function from Eq. (11).However, in reality, the short-time work time series is not perfectly uncorrelated, rendering the expression invalid.Nevertheless, we find that by modifying our approximations based on the mean and variance of W τm directly calculated from the Langevin dynamics of the process, we can achieve good agreement with experimentally and numerically measured distributions, particularly for typical work fluctuations (see Eq.(A.40) of Appendix AIV).This is illustrated in Fig. 6(a) and  (b).We also observe that positive work fluctuations are significantly more favoured in the viscoelastic case as compared to the corresponding viscous limit.In Fig. 6 (c) and (d), we further compute the ratio of probabilities [R(W τm )] for positive and negative work fluctuations.While the ratio is not linear for all work values, it shows linear behaviour with a unit slope for small work values, deviating for larger ones.The deviations are greater in a viscoelastic bath compared to a viscous environment, which again substantiates stronger deviations from equilibrium.

Conclusions
In summary, we have shown that the viscoelasticity of the environment has a significant impact on the thermodynamics and fluctuations of a driven colloidal particle.Specifically, we have found a substantial increase (around 50%) in the mean injected power to the system when compared to the same system in a viscous bath under the same external driving strength, even though the steady-state variance of the position fluctuations gets nominally (around 10%) enhanced.Additionally, we have observed lower negative work fluctuations across a wide range of driving amplitudes.Since the injected power corresponds to the thermodynamic arrow of time (entropy production) in this process, these findings suggest an enhanced directionality in a driven process when it takes place in a viscoelastic bath rather than a viscous bath with the same zero-shear viscosity.
In light of recent work [78], our findings suggest that the influence of viscoelasticity on directionality and thermodynamics may extend beyond the specific experimental setup studied in this work, potentially affecting a wide range of driven processes in biological contexts.For example, it will be interesting to study whether the viscoelasticity of the medium affects the directionality of molecular motors studied in vitro.Our results also suggest that viscoelastic environments may provide a more controlled and predictable environment for driven processes, potentially leading to improved performance and reliability in practical applications, such as the assembly of nanoscale materials [101].We plan to address some of these issues in a future work.where x 0 (t) is the random position of the potential minimum at time t originated from the equation ẋ0 ϕ ext (t) is a zero-mean delta-correlated Gaussian noise with unit standard deviation, and A is the strength of that noise.Note that x 0 (t) is an Ornstein-Uhlenbeck (OU) process with correlation time τ e , with the correlation given by ⟨x 0 (t)x 0 (t ′ )⟩ = A τe exp − |t−t ′ | τe .η ′ (t) is the Gaussian-distributed stochastic noise representing the thermal agitation of the fluid molecules.To satisfy the fluctuation-dissipation theorem (FDT) in equilibrium, the correlation of the noise is related to the memory kernel Γ(t) as ⟨η where k B is the Boltzmann constant and T is the temperature of the bath.As a viscoelastic model, we consider Jeffrey's fluid model where the memory Kernel is of the kind: where τ is the relaxation time of the fluid; γ 0 and γ are the solvent and the polymer contributions respectively.The thermal noise η ′ (t) can be represented as the sum of two zero-mean independent Gaussian processes as η ′ (t) = η ′ 0 (t)+η ′ 1 (t), where the correlations of these two processes are given by ⟨η ′ 0 (t)η ′ 0 (t τ respectively with γ 0 = 6πa 0 µ s and γ = 6πa 0 µ p .µ p and µ s are the viscosities of the polymer and the solvent, respectively.At the overdamped limit (where the inertial effect is negligible), after carrying out an integration by parts, Eq. (A.1) can be written as (A.4) We define η 0 (t) = η ′ 0 (t)/γ 0 and η 1 (t) = η ′ 1 (t)/γ 0 .Now, in order to model the system as Markovian we introduce an auxiliary variable and combining Eqns.(A.2), (A.4) and (A.5) we can write in the matrix form, where ϕ 0 , ϕ 1 and ϕ ext are Gaussian-distributed white noises with mean zero and unit standard deviation.
we can rewrite the Eq.(A.6) as the following: where D f = 1 2 DD T .The variance of x(t) is given by σ 11 and after solving the stationary condition, we get the following expression: .
(A.10) Interestingly, Var(x) shows non-monotonic dependence (Fig. 2(e) of main text) on the relaxation timescale of the fluid (τ ) with the minima at τ = τ min such that, It is clear that τ min has a non-trivial dependence on the other timescales of the system and is independent of the strength of external perturbation (A).
The expression for the variance of P (x) for a Newtonian fluid can be easily calculated by considering the limit τ → 0 in σ 11 , which becomes Note that this expression is similar to that obtained in an active viscous bath given in Ref. [59].The position auto-correlation function (ACF x ) of the particle is C 11 (t), i.e., ⟨x(t)x(t ′ )⟩ = C 11 (t − t ′ ), that can be easily calculated from Eq. (A.8).The meansquare displacement (MSD) of the particle for our system is related to the ACF x as

II. Work fluctuations
The stochastic mechanical work by the external force over a time τ m is defined as the following [77,6,65,66,63]: and the mean of work is given by, Considering c = x 0 (t) 0 0 and multiplying with Eq. (A.7) from the left we get As, ⟨x 0 (t)ϕ 0 (t)⟩ = 0 and ⟨x(t)x 0 (t)⟩ = σ 13 , ⟨X(t)x 0 (t)⟩ = σ  The relative viscosity µ r is the ratio of corresponding viscosities of the polymer (µ p ) and the solvent(µ s ). a 0 denotes the radius of the trapped particle.Now Eq.D.3 can be substituted in Eq.D.1 to get the phase response of the particle as: (D.4) As described in Ref. [71], the trapped microparticle embedded in the viscoelastic fluid is perturbed with a square wave modulation of frequency 3 Hz.The phase response of the particle to the odd harmonics of the square wave modulation is obtained experimentally and fitted with Eq.D.   semiconductor laser (Lasever, Max power 500 mW) of wavelength 1064 nm is tightly focused through the objective lens on the sample to trap a micro-particle dispersed in it.The trapping beam is passed through an acousto-optic modulator (AOM) placed at a plane conjugate to the objective lens.We modulate the beam by using the firstorder diffracted beam off the AOM with a computer-generated signal fed to the AOM controller via a data acquisition card (which acts as a signal generator here).We employ a second low-power laser beam (stationary and copropagating with the trapping beam) of wavelength 785 nm to track the position of the particle, which we determine from the back-scattered light incident on a balanced detection system [105], constructed using two high gain-bandwidth photo-detectors (Thorlabs PDA100A2) placed orthogonal to each other.The displacement data of the particle is recorded into a computer from the balanced detection signal via a data acquisition card (NI DAQ) at a spatio-temporal resolution of 1 nm-10kHz.

Figure 1 .
Figure 1.(a) Schematic of a spherical particle trapped in a viscoelastic bath by optical potential whose mean position (x 0 ) is externally modulated by Ornstein-Uhlenbeck noise.(b) Typical trajectories of the particle in (x, x 0 ) plane for different strengths of the external noise (Gray dots) with corresponding non-zero probability flux (Red arrows).

Figure 2 .
Figure 2. (a) Mean squared displacement (MSD) of the probe particle in the viscoelastic fluid.(b) Comparison of the MSD of the particle between viscous and viscoelastic fluid for θ = 0.56.The solid lines denote the theoretical predictions of all the cases.(c) Probability distribution of position fluctuations of the trapped particle in the viscoelastic fluid plotted for different strengths of the external noise.(d) The probability distribution of position fluctuations of the trapped particle in the viscoelastic and in viscous fluid plotted for a fixed strength of the external noise.θ = 0.56.(e) Theoretical estimation of the variance of the displacement of the trapped particle at different noise strengths plotted against the relaxation timescale of the fluid (τ ).The dash-dotted line denotes the timescale of the fluid used in the experiment.(Inset) The nonmonotonic behaviour of the variance with τ .

Figure 3 .
Figure 3. (a) Rate of mean work done by the external modulation on the trapped Brownian particle in viscoelastic bath and viscous bath ( τ → 0) plotted against the strength of the external noise.The solid lines denote the theoretical predictions.(b) Variation of work with the relaxation timescale of the medium plotted for different strengths of the external noise along with experimental points.The dash-dotted line denotes the timescale of the fluid used in the experiment.

Figure 4 .
Figure 4. (a)-(d) Probability density function of single-step work done (calculated from experimental data) by the external modulation on the trapped particle in the viscoelastic bath plotted for different strengths of the external noise.Corresponding probability ratios are plotted in (e)-(h).(i)-(l) The numerically calculated probability density functions of single-step work done are shown for different strengths of the modulation.(m)-(n) Numerically estimated probability ratios for the corresponding cases.Theoretical estimations in all the cases are represented by solid red lines.

Figure 5 .
Figure 5. Variation of β ws with τ .(Inset) β ws shows extremely slow variation with θ for both viscous and viscoelastic baths.

Figure 6 .
Figure 6.(a) Probability density function of work done (calculated from experimental data) by the external modulation on the trapped particle in the viscoelastic bath plotted for different strengths of the external noise.Theoretical estimations in all the cases are represented by solid lines.(b) Probability of positive fluctuations of work are enhanced for the viscoelastic bath as seen from the comparison of the same for the viscous bath.Theoretical estimations (Eq.(A.40) of Appendix IV) in all the cases are represented by solid lines.(c) R(W τm ) (estimated from the experimental data) plotted as a function of W τm for different strengths of the external modulation when the particle is in the viscoelastic environment.(d) R(W τm ) for the viscoelastic bath is found to be deviating from the usual work fluctuation theorem (black dashed line) with smaller external noise than the same for the viscous bath.

Figure B1 .
Figure B1.Ratio of probabilities corresponding to the positive and negative work fluctuations is plotted for different strengths of external noise in the viscoelastic (a) and viscous (b) bath, respectively.(c)Plot of 2β w as a function of θ.
4 to estimate the relaxation timescale (τ ), and relative viscosity (µ r ) of the fluid as shown in Fig.1(b) of the main text.

Figure D1 .
Figure D1.Phase response of the trapped particle for different frequencies and the corresponding fit for the phase-sensitive active micro-rheology [71].